The existence of periodic solutions for the third-order differential equation x¨˙+ω2x˙=μF(x,x˙,x¨) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if F(x,x˙,x¨) has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001)

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The aim of this paper is investigating the effect of vaccinating on the behavior of an infectious disease by study the effect on the basic reproduction number (R 0 (p)), in a population where two different categories of infected (also susceptible) individuals exist, the differences between these two infected (also susceptible) categories are the rates of infection transmission and recovery (for susceptible incidence rate) and vaccination program is carrying out by vaccinating two different percentages of these two different categories of susceptible individuals. To study the behavior of infectious disease R 0 (p), is computed and the relationship between R 0 (p) and the existence, stability and bifurcation of equilibriums is investigated. Also conditions that lead to removing the infection are obtained.

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